Chebyshevʼs bias in Galois extensions of global function fields

Chebyshev’s bias in Galois extensions of global function fields

Article history: Received 18 November 2009 Revised 6 January 2011 Accepted 15 March 2011 Available online xxxx Communicated by David Goss MSC: primary 11N05 secondary 11M38, 11G05

Skolem Density Problems over Large Galois Extensions of Global Fields

Let K be a global eld, V an innnite proper subset of the set of all primes of K, and S a nite subset of V. Denote the maximal Galois extension of K in which each p 2 S totally splits by K tot;S. Let M be an algebraic extension of K. A data for an (S; V)-Skolem density problem for M consists of a nite subset T of V containing S, polynomials f 1 ; : : : ; f m 2 ~ KX 1 ; : : : ; X n ] satisfying j...

Galois Extensions of Hilbertian Fields

We prove the following result: Theorem. Let K be a countable Hilbertian field, S a finite set of local primes of K, and e ≥ 0 an integer. Then, for almost all ∈ G(K)e, the field Ks[ ] ∩Ktot,S is PSC. Here a local prime is an equivalent class p of absolute values of K whose completion is a local field, K̂p. Then Kp = Ks ∩ K̂p and Ktot,S = T p∈S T σ∈G(K) K σ p . G(K) stands for the absolute Galois ...

UNRAMIFIED EXTENSIONS AND GEOMETRIC Zp-EXTENSIONS OF GLOBAL FUNCTION FIELDS

We study on finite unramified extensions of global function fields (function fields of one valuable over a finite field). We show two results. One is an extension of Perret’s result about the ideal class group problem. Another is a construction of a geometric Zp-extension which has a certain property.

Compositum of Galois Extensions of Hilbertian Fields